A branch and bound algorithm for minimizing makespan on a single machine with unequal release times under learning effect and deteriorating jobs

Abstract

We present a single-machine problem with the unequal release times under learning effect and deteriorating jobs when the objective is minimizing the makespan. In this study, we introduced a scheduling model with unequal release times in which both job deterioration and learning exist simultaneously. By the effects of learning and deterioration, we mean that the processing time of a job is defined by increasing function of its execution start time and position in the sequence. A branch-and-bound algorithm incorporating with several dominance properties and lower bounds is developed to derive the optimal solution. A heuristic algorithm is proposed to obtain a near-optimal solution. The computational experiments show that the branch-and-bound algorithm can solve instances up to 30 jobs, and the average error percentage of the proposed heuristic is less than 0.16%.

Introduction

This paper addresses a single-machine problem under learning effect and deteriorating jobs when the objective is minimizing the makespan when the unequal release times are available. In the classical scheduling theory, job processing times are considered to be constant. In practice, however, we often encounter setting in which processing times increase or decrease over time [1]. There is a growing interest in the literature to study scheduling problems of deterioration jobs, i.e., jobs whose processing times are increasing functions of their starting times. Gupta and Gupta [2] provided an example of steel rolling miles where the temperate of an ingot, while waiting to enter the rolling machine, drops below a certain level, required the ingot to be reheated before rolling. Furthermore, Kunnathur and Gupta [3] and Mosheiov [4], [5] presented several real-life situations for deteriorating jobs. They contain shops with deteriorating machines, delay of maintenance or cleaning, fire fighting, hospital emergency wards, the control of queues and communication systems, etc., in which any delay in processing a job may result in an increasing effort to accomplish the job.

On the other hand, some researchers assumed that the time of task is independent from learning of worker(s) for repetition jobs. The impact of the learning effect on production issues was first discussed by Wright [6], who observed that learning may decrease the processing times of production tasks in the aircraft industry. The observation by OkoŁowski and Gawiejnowicz [7] was later confirmed by many empirical studies saying that the knowledge of a learning curve during the planning process may result in cost savings in manufacturing [8], industrial production [9] and software engineering [10]. However, in many realistic settings, workstations improve continuously as a result of repeating the same or similar activities. Thus, the processing time of a job is shorter if it is scheduled later, rather than in the sequence [1]. Mosheiov [11] determined that this phenomenon is known in the literature as a “learning effect”.

Recently, the effects of learning and deterioration are simultaneously considered in some scheduling problems because the phenomena can be found in many real-life situations. For example, Wang [12] pointed out that as the manufacturing environment becomes increasingly competitive, firms are moving towards shorter production runs and frequent product changes in order to offer faster services and provide customers with greater product varieties. The learning and forgetting that workers undergo in this environment have thus become increasingly important as workers tend to spend more time in rotating among tasks and responsibilities prior to becoming fully proficient in carrying out their operations. These workers are often interrupted by product and process changes that cause deterioration in their operational performance. For this situation, considering both the job deterioration and learning effect in job scheduling is both necessary and reasonable [13]. As a second example, Toksari and Güner [14] considered jewelry sector because both skill of workers and used material structure are very important. If the temperate of the material used for jewelry production, while waiting to enter the rolling machine, drops below a certain level, it must be reheated before rolling. Namely, material has deteriorated in the course of time. Furthermore, some researchers are worked on scheduling problems with both learning effect and deterioration jobs. To the best of our knowledge, Wang et al. [15] proposed the solutions with polynomial time for single machine scheduling problems under deteriorating jobs and learning effect. Ghodratnama et al. [16] worked on minimizing the sum of the weighted jobs completion, minimizing the sum of the weighted delay times, and maximizing the sum of the job values in makespan with maintenance, job deterioration and learning effect. They used the simulated annealing approach. Xingong and Guangle [17] present single-machine group scheduling problems with deteriorating jobs and learning effect. Wang and Guo [18] investigated a due date assignment problem with learning effect and deteriorating jobs. Huang et al. [19] used exponential learning effect and time dependent deterioration for some single machine scheduling problems. Toksari and Güner [20] worked on parallel machine the common due-date early/tardy scheduling problem under the effects of time-dependent learning and linear and nonlinear deterioration. Wang [21] showed that even with the introduction of learning effect and deteriorating jobs to job processing times, single machine makespan and sum of completion times (square) minimization problems remain polynomially solvable, respectively. But for the following objective functions: the weighted sum of completion times and the maximum lateness, this paper proves that the WSPT rule and the EDD rule can construct the optimal sequence under some special cases, respectively. Sun [22] showed that the problems of makespan, total completion time and the sum of the quadratic job completion times remain polynomially solvable, respectively. In addition, he showed that the problems to minimize total weighted completion time and maximum lateness are polynomially solvable under certain conditions. Wang and Liu [23] worked on two-machine flow shop problem with effects of deterioration and learning. Toksari and Güner [24] investigated parallel machine earliness/tardiness scheduling problem under the effects of position based learning and linear/nonlinear deterioration. Toksari et al. [25] used the effects of nonlinear deterioration and time-dependent learning for some single machine scheduling problems. Cheng et al. [26] introduced a scheduling model in which both job deterioration and learning exist simultaneously. They worked some single machine problems and m-machine permutation flowshop problems. For makespan minimization problem, Wang and Cheng [27] showed that the schedule produced by the largest growth rate rule is unbounded for their model, although it is an optimal solution for the scheduling problem with deteriorating jobs and no learning. They then consider three special cases of the problem, each corresponding to a specific practical scheduling scenario. Based on the derived optimal properties, they developed an optimal algorithm for each of these cases and considered a relaxed model of the second special case, and present a heuristic and analyze its worst-case performance bound. Wang [12] proposed the introduction of learning effect and deteriorating jobs to job processing times; single-machine makespan and sum of completion times (square) minimization problems remain polynomially solvable, respectively.

However, researches are relatively limited on learning and release times. Wu and Liu [28] proposed a branch-and-bound algorithm and three two-stage heuristic algorithms for the problem which is minimizing the makespan on a single machine with learning and unequal release times. Lee et al. [29] further provided exact and heuristic algorithms for the same problem. Eren [30] investigated a single-machine scheduling problem with unequal release dates and a position-based learning where the objective is to minimize the total weighted completion time. He developed a nonlinear mathematical programming model for the problem. In this paper, we introduced a scheduling model with unequal release times in which both job deterioration and learning exist simultaneously.

The rest of the paper is organized as follows: in Section 2, we will present formulation of the problem under study. In Section 3, branch and bound algorithm and a heuristic algorithm are developed after some dominance properties and lower bounds are explained. The results of computational experiment are given in Section 4. The conclusions of the research are summarized in the last section.